What is the definition of each of these neural network cell types?
The diagram is nice-ish-looking, and it does lack a convenient list of node type descriptions.  There is some description in their Zoo Flashcards page. Although the zoo is not a clean presentation in every respect, it is quite instructive in several ways. These are of particular interest. 


The symbolic shapes and colors visually display the relationship between cell types and the types of networks that employ them, which is quite instructive as an overview.
The use of the term cell, as opposed to neuron, acknowledges the fundamental difference between current artificial network cells and the amazingly intricate and self-modifying nature of biological neurons.


What is still missing is a concise list of each cell type and its purpose for existing in the contexts of its use.  This may be useful.


Input Cell &mdash; Provides a point of input and often uses an identity activation function (the output = the input)
Back-fed Input Cell &mdash; Provides a point where feedback is deliberately introduced
Noisy Input Cell &mdash; Provides a point where chaotic (pseudo-random), thermal, or quantum noise can be introduced
Hidden Cell  &mdash; A cell that attenuates (applies a parameterized weight to) the input from all or a select range of activation function outputs from the cells of the immediately prior layer
Probabilistic Hidden Cell &mdash; Applies a radial basis function to the difference between the test case and the cell's mean1
Spiking Hidden Cell &mdash; Employs a spiking model of activation rather than a perceptron model, working toward a more accurate simulation of biological neurons at the cell level
Output Cell &mdash; Provides a point of output and applies the range (min and max) or ordinality (binary, category index)
Match Input Output Cell &mdash; Used in auto-encoders to converge on the identity function from network input to network output, usually through a narrowed data path through which features are extracted after convergence
Recurrent Cell &mdash; Used in RNN designs such as recurrent artificial networks as a method of applying historic signal state to improve convergence where the time domain is indicated as an important dimension of learning 2
Memory Cell &mdash; Used in LSTM (long short term memory) designs as a method of applying historic signal state to improve convergence where the time domain is indicated as an important dimension of learning 2
Different Memory Cell &mdash; Used in GRU (gated recurrent unit) designs as a method of applying historic signal state to improve convergence where the time domain is indicated as an important dimension of learning 2
Convolution Cell &mdash; Passes the result of a convolution kernel multiplied to a relative range of the cells of the immediately previous layer to the next layer 
Pool Cell &mdash; Aggregates (usually arithmatically) the results of the cells of a convolution layer


A table of cell types with the features for each might be a next step in presenting the cell types in a comparative summary.  The table would particularly helpful if it had sort up and down gadgets in the header row for each column.

Footnotes

[1] Typically $\varphi(\mathbf{x}) = \sum_{i=1}^I a_i e^{\left[ -\beta \left \Vert \mathbf{x} - \mathbf{c}_i  \right \Vert ^2 \right]}$ where c is the center value of the cell

[2] These cell types integrate historical influence into the signal path at the cellular level, introducing statefulness beyond the learning parameters and network hyper-parameters, providing the potential of Turing completeness to the network after training.